Transcription of 8.2 Quadratic Forms Example 1 - NCU
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8.2 Quadratic Forms Example 1 - NCU
(x1; x2) = 8x21 4x1x2+ 5x22 Determinewhetherq(0;0) is matrixtechniqueRewriteq("x1x2#) = 8x21 4x1x2+ 5x22="x1x2# "8x1 2x2 2x1+ 5x2#Notethatwe splitthecontribution 4x1x2equallyamongthetwo succinctly, we canwriteq(~x) =~x A~x;where A="8 2 25#1orq(~x) =~xTA~xThematrixAis symmetricby ,thereis an orthonormaleigenbasis~v1; ~v2forA. We nd~v1=1p5"2 1#; ~v2=1p5"12#withassociatedeigenvalues 1= 9 and 2= ~x=c1~v1+c2~v2, we canexpressthevalueof thefunctionas follows:q(~x) =~x A~x= (c1~v1+c2~v2) (c1 1~v1+c2 2~v2)= 1c21+ 2c22= 9c21+ 4c22 Therefore,q(~x)>0 for all nonzero~ (0;0) =0 is theglobalminimumof functionq(x1; x2; : : : ; xn) fromRntoRiscalleda quadraticformif it is a linear combina-tionof functionsof theformxixj. A quadraticformcanbe writtenasq(~x) =~x A~x=~xTA~xfor a symmetricn (x1; x2; x3) = 9x21+7x22+3x23 2x1x2+4x1x3 6x2x3 Finda symmetricmatrixAsuchthatq(~x) =~x A~xfor all~ Example1, we letaii= (coe cientofx2i),aij=12(coe cientofxixj), ifi6= ,A=2649 12 17 32 333752 Changeof Variablesin a quadraticformq(~x) =~x A~xfromRntoR.
8.2 Quadratic Forms Example 1 Consider the function q(x1;x2)=8x21 4x1x2 +5x22 Determine whether q(0;0) is the global mini-mum. Solution based on matrix technique Rewrite
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